Коэффициенты масштабирования, угловые соотношения и треугольники
Введено| English | Значение |
|---|---|
| perpendicular height/ˌpɜːpənˈdɪkjʊlə haɪt/ | перпендикулярная высота |
| similarity/ˌsɪmɪˈlærɪti/ | подобие |
A decision before an answer
- A cube with doubled edge has eight times the volume. Doubling its visible width is not doubling every measure.
- Your goal: Apply length, area and volume scaling to similar figures.
Read the relationship
- For similar figures with length scale k, corresponding lengths scale by k, areas by k² and volumes by k³. These laws apply when all corresponding dimensions change together. A cylinder with only radius doubled and fixed height has four times the volume, not eight times. State which dimensions are scaled before choosing the power.
- Use parallel-line angles and triangle sums to find unknowns.
Similar solids have length scale 3. Volume scale:
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Volume uses k³.
Use the defining rule
- Use the correct geometric measure: triangle area is half base times perpendicular height; cylinder volume is base area times height. Surface area adds exposed faces, while volume measures enclosed space. A slant length is not automatically the perpendicular height required by an area formula. Attach square or cubic units to the result.
- Distinguish similarity from congruence and adequate information.
Triangle angles 35° and 85° leave:
Difficulty 1/5 · 100 coins + a small bonus on first completion today
180-35-85=60.
Check the conditions
- Vertical angles are equal. For parallel lines cut by a transversal, corresponding and alternate interior angles are equal, and same-side interior angles sum to 180°. A triangle’s interior angles sum to 180°; its exterior angle equals the sum of the two remote interior angles. These rules depend on stated or marked parallelism, not the drawing’s appearance.
- Distinguish similarity from congruence and adequate information.
Two similar solids have edge ratio 2:3, so surface-area ratio is 4:9 and volume ratio 8:27. A triangular garden with base 12 m and perpendicular height 5 m has area 30 m². If a triangle has interior angles 48° and 67°, its third is 65° and the adjacent exterior angle is 115°=48°+67°.
Area scale for length scale 4 is ____.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Area uses 4².
Apply the task format
- Similar triangles share corresponding angles and proportional corresponding sides; congruent triangles also have equal corresponding lengths. Two angles establish similarity but not size equality. Maintain correspondence when writing a ratio. A side–side–angle arrangement does not universally guarantee congruence, whereas side–angle–side includes the angle between the two sides.
- Distinguish similarity from congruence and adequate information.
Use k² or k³ only when the entire shape is scaled; do not infer parallel lines or a perpendicular height from an unmarked sketch.
Какой ответ подходит к данному случаю?
Применяйте масштабирование длин, площадей и объемов для подобных фигур ⟦number⟧
Two equal corresponding angles prove equal triangle sizes.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
They prove similarity, not equal scale.
Keep the distinctions
- similarity 相似 — Matching shape with proportional corresponding lengths.
- perpendicular height 垂直高度 — The shortest perpendicular distance used with a chosen base.
- Apply length, area and volume scaling to similar figures.
- Use parallel-line angles and triangle sums to find unknowns.
- Distinguish similarity from congruence and adequate information.
Match each term with its precise meaning in this lesson.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
Difficulty 3/5 · 200 coins + a small bonus on first completion today
The order follows the stated process; check each stage before the next.